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Â Now, in the previous analysis we considered the non-interacting case.

Â That is, we neglected any interactions between absorbed molecules.

Â Now, this is an adequate description when the surface coverage is low.

Â However, when surface coverage becomes high, interaction between

Â absorbed molecules becomes important and this leads to correlated surface coverage.

Â Now, this is analogous to the spacial correlations that are found

Â in the Eisen model.

Â Let's adopt a simple model for the absorption energy on a lattice.

Â In this expression, ni stands for the site occupancy.

Â Now ni can take the value 0 or 1, 0 if it is unoccupied, 1 if it is occupied.

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Now K in this expression describes the coupling constant

Â to describe the interactions between neighboring adsorbed molecules.

Â An F bind is the free energy associated with binding of the absorbed molecule.

Â Now in this system, we need to take into account

Â the changes in the particular number of absorbed molecules.

Â Now what do you think?

Â Which ensemble should we pick?

Â Now the most convenient ensemble to pick is the grand canonical ensemble.

Â The grind canonical quotation function can be written in a standard way.

Â Now rather than carrying out a more detailed analysis of this equation

Â we will take a simpler approach and cast this in the language of the Eisen model.

Â What we need to do to do this Well, we need a change of variables where we can

Â relate the occupancy number NI to the spin variable SI by a linear transformation.

Â Now the simple mapping, maps the occupancy variable to a spin value of one.

Â When occupied, and a spin value of minus one when not occupied.

Â Now, this variable transformation allows us to recast the problem exactly

Â in the Eisen model.

Â In this expression, Z stands for

Â the coordination number of the lattice given by the number of nearest neighbors.

Â Now we have to remember to not double count, so

Â we need to divide this number by a factor of two.

Â So, for a two dimensional square lattice, the Z value takes two.

Â Therefore, with this exact map mapping of the current problem to the Ising model.

Â We can now evaluate all of these things in an analogous way to what we found in

Â the Ising model.

Â Therefore, we can exactly map our current problem

Â onto the Ising model with a suitable set of parameters.

Â The following parameter mapping holds for

Â the absorption case to the standardizing model.

Â Owing to the exact mapping between our current and the icing model,

Â we can note a few identical physical phenomena that arises in both systems.

Â Adsorption with interactions, exhibit of phase transition.

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Well, this phase transition occurs when the coupling constant is sufficiently

Â large and more importantly, the lattice is not one dimensional.

Â Now, adsorption exhibits hysteresis in the surface coverage

Â versus gas pressure associated with the phase transition.

Â Now, the surface coverage is not completely uncorrelated.

Â Rather, the adsorbed molecules exhibit correlated fluctuations

Â due to favorable interactions.

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And finally, the critical point in the adsorption isotherm is associated with

Â a divergence of the correlation length of these spatial fluctuations.

Â Now, to summarize, in this video we learned a lot about surface adsorption.

Â We discuss two specific cases of surface absorption

Â without any surface interactions.

Â Now this is the case of localized absorption and mobile absorption.

Â We derived expressions for

Â the surface coverage in these cases under constant temperature.

Â And then we proceeded to describe a picture for

Â surface absorption that accounts for interacting aborb molecules.

Â We showed the equivalence of this problem to a problem model

Â using a suitable change of variables.

Â Now this system exhibits a phase transition under sufficiently large

Â coupling constant and when the lattice is not one dimensional.

Â